Reducing noise in phased-array signals from receivers located at different locations

ABSTRACT

In a computerized method to reduce noise in phased-array signals from a set of receivers at different locations, time-series are received from the receivers, which time-series form phased-array signals. The time-series are ordered based on the different locations of the receivers and spatially phased series are obtained from the ordered time-series. Each of the spatially phased series obtained includes a series of signal values that are spatially phased. A noise component is identified in each of the spatially phased series obtained and removed from the spatially phased series to obtain denoised series. Related receiver systems and computer program products are also provided.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a continuation of U.S. patent application Ser. No.15/254,208 filed Sep. 1, 2016, the complete disclosure of which isexpressly incorporated herein by reference in its entirety for allpurposes.

BACKGROUND

The invention relates in general to computerized method to reduce noisein phased-array signals. It can in particular be applied to the fieldsof radio interferometry (to clean a sky image), magnetic resonanceimaging or ultrasound imaging and, more generally, sensor networks.

A phased-array is an array of sensors (or, in certain applications,antennas) which preserves a specific direction-based relationshipbetween the phases of the respective signals. Phased-arrays are used innumerous areas such as ultrasound, magnetic resonance imaging, radioastronomy, optics and many others. Typically, the thermal noise sensedby the sensors has high power relative to weak incoming signals. Currentsolutions to overcome this noise usually rely on the use of asufficiently large number of sensors and extend the observation time soas to find the true signal hidden in the noise.

The following references will later be referred to:

-   Jonathan Gillard. Cadzows basic algorithm, alternating projections    and singular spectrum analysis. Statistics and its interface, 3    (3):335-343, 2010.-   Nina Golyandina and Anatoly Zhigljaysky. Singular Spectrum Analysis    for time series. Springer Science & Business Media, 2013.-   Ying, Leslie and Liang, Zhi-Pei. Parallel MRI Using Phased Array    Coils. Signal Processing Magazine, IEEE, 27 (4):90-98, July 2010.

SUMMARY

According to a first aspect, the present invention is embodied as acomputerized method to reduce noise in phased-array signals from a setof receivers at different locations. Time-series are received from thereceivers, which time-series form phased-array signals. The time-seriesare ordered based on the different locations of the receivers andspatially phased series are obtained from the ordered time-series. Eachof the spatially phased series obtained comprises a series of signalvalues that are spatially phased. A noise component is identified ineach of the spatially phased series obtained and removed from thespatially phased series to obtain denoised series.

The spatially phased series as obtained before denoising from theordered time-series, may comprise a series of signal values that areboth spatially phased and time shifted. I.e., the ordering of thetime-series may comprise selecting time-series elements corresponding totwo or more time instances and ordering the time-series elementsselected based on the different locations of the receivers. Thus, thespatially phased series obtained may comprise a series of signal valuesthat correspond to signal sensed at two or more time instances.

In variants, the spatially phased series obtained for each time-instancecomprise a series of signal values that are only spatially phased, wherethe phase difference solely occurs due to the different locationsbetween the receivers. That is, the ordering of the time-series isperformed for each time-instance of the time-series received, based onthe different locations of the receivers, so that each of the spatiallyphased series obtained for each time-instance comprises a series ofsignal values that correspond to signal sensed at a same time instance.

The ordering of the time-series is preferably based on distances betweenthe different locations of the receivers. For example, it may beperformed by identifying a sequence of receivers, wherein an n+1^(th)receiver of the sequence is the closest receiver from an n^(th) receiverof the sequence, so as to minimize distances between locationscorresponding to contiguous pairs of elements in each of the spatiallyphased series.

In embodiments, the noise component is identified and removed accordingto steps of a singular spectrum analysis. Preferably yet, such stepsinclude a step of eliminating frequencies above a given threshold insingular-vectors of one of two disjoint sets of singular triples thatcomprises highest largest singular-values, prior to reconstruct anapproximate de-noised, spatially phased series.

The receivers may for instance be radio astronomy antennas grouped instations, radiofrequency coils of a magnetic resonance imaging system,transducers of an ultrasound apparatus, or, more generally, sensors of asensor network. The receivers may for instance be devices of anInternet-of-Things network.

According to another aspect, the invention is embodied as a receiversystem. The system comprises a set of receivers at different locations,wherein each of the receivers is configured to convert signals itreceives into time-series. The system also comprises one or more noisereduction units connected to the receivers. Each noise reduction unit isconfigured to receive time-series from receivers and order thetime-series based on the different locations of such receivers, toobtain spatially phased series. Each noise reduction unit is furtherconfigured to identify a noise component in each of the spatially phasedseries obtained and remove the identified noise component therefrom,consistently with principles of the present methods.

In embodiments, the receiver system is a radio interferometry system andthe receivers are radio astronomy antennas that are grouped in stationsin the system. Preferably, the system comprises at least one noisereduction unit per station. In variants, the receiver system can be amagnetic resonance imaging system (in which case the receivers areradiofrequency coils) or an ultrasound apparatus (in which case thereceivers are transducers).

According to a final aspect, the invention can be embodied as a computerprogram product for reducing noise in phased-array signals from a set ofreceivers. The computer program product comprises a computer readablestorage medium having program instructions embodied therewith, theprogram instructions being executable by a computerized system to causeto take steps according to the present methods.

Devices, apparatuses, systems, methods and computer program productsembodying the present invention will now be described, by way ofnon-limiting examples, and in reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flowchart illustrating high-level steps of a method forreducing noise in phased-array signals received from a set of receivers,according to embodiments;

FIG. 2 is a flowchart illustrating steps of a modified singular spectrumanalysis, which can be used as part of the method of FIG. 1, inembodiments;

FIG. 3 is a flowchart illustrating additional steps to obtain beamformedoutputs and correlation values, as in embodiments directed tointerferometry applications;

FIG. 4 schematically depicts a radio interferometry system, whosereceivers are antennas grouped in stations, according to embodiments;

FIG. 5 schematically represents a magnetic resonance imaging system,according to embodiments;

FIG. 6 schematically depicts an ultrasound apparatus, according toembodiments;

FIGS. 7A and 7B are plots of an example of signal that includes thermalnoise, the corresponding true signal (without thermal noise) and anestimate of the true signal reconstructed according to an embodiment ofthe present methods;

FIGS. 8A and 8B are density plots that represent differences between, onthe one hand, a true signal (without thermal noise) and an estimatethereof as reconstructed using a conventional, prior art method (FIG.8A) and, on the other hand, an estimate as reconstructed according to anembodiment of the present methods (FIG. 8B); and

FIG. 9 schematically represents a general purpose computerized system,which can for example be configured as a de-noising unit, forimplementing one or more method steps as involved in embodiments.

The accompanying drawings show simplified representations of devices,systems or apparatuses according to embodiments, or parts thereof.Similar or functionally similar elements in the figures have beenallocated the same numeral references, unless otherwise indicated.

DETAILED DESCRIPTION

The following description is structured as follows. First, generalembodiments and high-level variants are described (sect. 1). Section 2briefly exposes a modified singular spectrum analysis (SSA) techniqueused to denoise phased-array signals. Section 3 describes properties ofembodiments of the invention. Section 4 addresses particularembodiments, in detail. The last section (sect. 5) is directed totechnical implementation details of some embodiments.

1. General Embodiments and High-Level Variants

In reference to FIGS. 1, and 4-6, an aspect of the invention is firstdescribed, which concerns a computerized method for reducing noise inphased-array signals received from a set of receivers 410, 510, 610 thatare located at different locations.

The method can for instance be implemented in a general-purpose ordedicated computerized system, in a centralized fashion or in a partlyor fully delocalized manner. According to this method, time-series arereceived (step S11, FIG. 1) from the receivers 410, 510, 610. Thereceivers may notably be radio astronomy antennas of a radiointerferometry system, radiofrequency coils of a magnetic resonanceimaging (MRI) system, transducers of an ultrasound apparatus, or, moregenerally, sensors of a sensor network, as discussed later in referenceto FIGS. 4-6. Each receiver senses a signal at each time instance and isable to produce a sensed signal value for each time instance, such thattime series can be received from the receivers. Altogether, thetime-series form phased-array signals. Phased-array signals are knownper se.

According to the present methods, the time-series are re-ordered.Namely, the time-series received S11 are ordered S12 based on thedifferent locations of the receivers 410, 510, 610. This makes itpossible to obtain S13 spatially phased series, which comprise, each, aseries of signal values (i.e., elements) that are spatially phased. Thatis, elements in each series are spatially phased according to an outcomeof step S12. The phase difference occurs due to the different locationsof receivers. Yet, such elements may further be explicitly time shifted,i.e., by explicitly incorporating a time shift in the signal values (orelements) of each series, as explained later in detail.

Next, a noise component is identified S14 in each of the spatiallyphased series obtained S13, so that the identified noise components canbe removed S15 from the spatially phased series. Eventually, denoisedseries are obtained S16, which are spatially phased, from whichestimates of the true signal can be reconstructed.

The apparent order of the above steps does not necessarily represent theorder in which these steps will actually be performed, in operation. Forinstance, one may preferably want to perform the ordering S12 during abuilt-time, e.g., once for all, so that the spatially phased seriessubsequently obtained S13 may all obey a same order. Of course, theunderlying system may be configured to allow some re-ordering, fromtimes to times.

The above solution allows the noise in phased-arrays to be significantlyand faithfully reduced, even in presence of very high instrumentalnoise, as present inventors have observed. Examples will be giventhroughout the description. While prior art denoising solutions forphased-array signals typically act on a large set of sensors and extendthe observation time, the present solution de-noises the time-seriesdirectly from the receivers, thus increasing substantially the accuracyof results. Conversely, the present solution can also enable thereduction of the number of receivers and/or measurements required, thuspotentially reducing substantially the cost of building a phased-array.

Embodiments of the invention find multiple applications, e.g., inmagnetic resonance or ultrasound imaging (and more generally medicalimaging), sensor networks, radio astronomy and interferometry, opticalphased arrays, microphone arrays, etc., allowing various signal sources(stars and other wave sources, sounds, diseases, organs, etc.) to beobserved. The present solution can also be applied to cognitivecomputing and internet of things technologies. The denoising methodsdiscussed herein can further be thought of as unsupervised machinelearning techniques, which may lead to potentially vastly improved data,learning images, features, etc. The present denoising techniques impactthe number and robustness of sensors (notably, cheaper sensors can beused), which may for instance operate in an internet of thingsframework.

In embodiments, the spatially phased series obtained at step S13 may beboth spatially phased and time shifted. I.e., the ordering S12 of thetime-series may be performed by selecting time-series elements thatcorrespond to two or more time instances and ordering the time-serieselements selected based on the different locations of the receivers.Thus, the spatially phased series obtained S13 comprise a series ofsignal values that correspond to signal sensed at two or more timeinstances.

In simpler variants, however, the spatially phased series obtained atstep S13 are only spatially phased, as exemplified below. I.e., thephase difference solely occurs due to the different locations betweenthe receivers. That is, the ordering S12 of the time-series is performedfor each time-instance of the time-series received, based on thedifferent locations of the receivers. Thus, each spatially phased seriesis obtained S13 for each time-instance and comprises a series of signalvalues that correspond to signal sensed at a same time instance. In thiscase, the ordering S12 may be performed according to a predeterminedorder of the receivers, e.g., by identifying a sequence of receiversbased on receivers' locations. Typically, a number of spatially phasedseries will subsequently be obtained according to that samepredetermined order. One will typically keep on using that same order,unless the system is re-parametrized, e.g., to take into account newreceivers, new locations, or any relevant change in the set ofreceivers. Only one spatially phased series is typically obtained ateach time instance.

The ordering of the time-series is preferably based on distances betweenthe different locations of the receivers 410, 510, 610. The ordering S12may for instance be performed by identifying a sequence of receivers, bysuccessively picking closest receivers, such that the n+1^(th) receiverof the sequence is the closest receiver from the n^(th) receiver of thesequence. This way, one minimizes distances between locationscorresponding to contiguous pairs of elements in each of the spatiallyphased series. In practice, such a sequence may be identified based onpair distances.

For example, the algorithm may proceed as follows. A first receiverlocation L₁ is picked-up, which is assumed to be at the origin andcorresponds to one particular receiver (conveniently called R₁) of a setof N receivers. Then, a closest receiver location L₂ is identified,which is distinct from L₁ and corresponds to a second receiver of theset, denoted by R₂. A next closest receiver location L₃ is thenidentified (distinct from each of L₁ and L₂), and so on. An order canthus be established, S12, based on physical locations (i.e.,geographical positions) of the receivers, which locations may typicallybe set in a 2D or 3D space.

Assume now that each receiver collects only one value at each timeinstance, for simplicity. The signals sensed by each receiver R_(i) atlocation L_(i) are converted into time-series c_(i)(t) and provided to asuitable computerized unit, e.g., a denoising unit (or denoiser forshort) 430, 530, 630 (FIGS. 4-6), for the latter to perform thefollowing operation.

For a given time instance, e.g., t=t_(j), a spatially phase series{tilde over (x)}_(j)=x(t_(j)) is obtained S13, such that {tilde over(x)}_(j)={c₁(t_(j)), c₂(t_(j)), . . . , c_(N)(t_(j))} and the sameordering is kept for each subsequent time series (unless, of course, thedenoiser is re-parameterized). In this example, the space-series arespatially-phased (due to the different locations of the receivers) butno additional time shift is imposed in the spatially phase series {tildeover (x)}_(j). I.e., two contiguous elements c_(k)(t_(j)),c_(k+1)(t_(j)) of the series correspond to signal sensed by two closestreceivers of the set {k, . . . , N} at a same time instance t_(j),according to the ordering performed at step S12. Other orderingstrategies may be implemented, which may be more or less sophisticated.In all cases, this ordering will be based on locations of the receivers.

However, as evoked above, additional time shift may be imposed in thespatially phase series {tilde over (x)}_(j) as obtained at the end ofstep S13. That is, the series obtained may include additional timeshifts, in addition to the spatial phase resulting from thelocation-based ordering. A simple example is one where time-series arebuffered for M successive time points and the space-time series obtainedare simple time shifts of the previous example, whereby {tilde over(x)}_(j)={c₁(t_(j)), c₂(t_(j−1)), . . . , c_(N)(t_(j−M))}. In thisexample, two contiguous elements c_(k)(t_(j),), c_(k+1)(t_(j′−1)) of theseries correspond to signal sensed by two closest receivers of the set{k, . . . , N} at two distinct (but contiguous) time instances. Ofcourse, more sophisticated algorithms may be involved, depending on theproblem to be solved. It may for instance be needed to impose particulartime shifts, due to large separation distances between the receivers andthe finite propagation speed of the signal meant to be detected, as in(radio) astronomy applications. In general, additional time shift may beimposed in the spatially phase series {tilde over (x)}_(j) as obtainedat the end of step S13 by selecting time-series elements that correspondto two or more different time instances and then ordering thetime-series elements selected, according to the locations of thereceivers.

In addition, if more than one values are to be collected by some or eachof the receivers at each time instance, then the problem can belinearized, by considering each receiver concerned as several receivers.In that respect, we note that different observations made by onereceiver at a same instant may correspond to different physicallocations within that receiver (as in multi-pixel sensors, whose pixelsare at different locations). In such cases, linearization is all themore appropriate.

As evoked earlier, the method may further comprises a step ofreconstructing S17 an estimate of a true signal according to thedenoised, spatially phased series obtained S16. This step need not belocally performed, e.g., by a denoiser 430, but may be performed byanother computerized unit 450 or module, which may collect denoisedseries as obtained at step S16, for one or more, or each time instance,to that aim.

Referring now to FIGS. 1 and 2, in embodiments, the identification S14and removal S15 of the noise components may advantageously be performedaccording to steps of a singular spectrum analysis (SSA) techniqueS21-S26. The basic SSA algorithm is known to involve four steps, namelyan embedding step, a Singular Value Decomposition (or SVD), a so-calledEigentriple grouping step, and a diagonal averaging.

General principles of the basic SSA are echoed in steps S21, S22, S23and S26 of FIG. 2, although such steps may typically need be adapted tothe context. In particular, the basic SSA analysis S21-S26 mayadvantageously include an additional step S24, which aims at eliminatingS24 highest frequencies. I.e., frequencies above a given threshold insingular-vectors of one of two disjoint sets of singular triples mayneed be discarded. The set of singular triples concerned is the set thatcomprises the lowest singular-values. This additional step S24 isperformed prior to reconstruct S26 approximate de-noised, spatiallyphased series. This step S24 impacts the subsequent step S25 (low-rankapproximation, FIG. 2), wherein a trajectory matrix of the true signalis approximated. Yet, step S25 may be regarded as being part of thefinal reconstruction step S26.

The additional step S24 is all the more advantageous when the noise isbelieved to be substantial, even much larger than the true signal so todistinguish noise and true signal easily. All this shall be explained indetail in the next sections. Yet, we note that the modified SSAtechniques used herein are optional, inasmuch as other denoisingtechniques may be contemplated, depending on the problem at stake, asthe skilled person will appreciate. Cadzow denoising, for instance,performs well when noise has low power. Basic SSA is also used todenoise high SNR signals. Note that the steps in FIG. 2 are notindependent from the general flowchart of FIG. 1. Rather, steps S14-S16of FIG. 1 may, in embodiments, be implemented as steps S21-26 of theparticular embodiment of FIG. 2.

Referring now to FIGS. 4-6, and according to another aspect, theinvention can be embodied as a receiver system 4-6. Such a systembasically comprises a set of receivers 410, 510, 610 (located atdifferent locations), wherein each of the receivers 410, 510, 610 isconfigured to convert signals it receives into time-series. The system4-6 further comprises one or more denoisers (or modules) 430, 530, 630,which are connected to the receivers 410, 510, 610. Each denoiser isconfigured to implement steps S11-S16 as described above in reference tothe present methods. Depending on the context, step S17 may be performedat the same denoisers 430, 530, 630 or modules that implement stepsS11-S16, or at one or more connected units 450, 550, 650 or modulesoperatively connected to the former.

One or more denoisers 430, 530, 630 may indeed be involved. For example,in radio interferometry applications (FIG. 4), it may be advantageous toprovide at least one denoiser 430 per station. In other applications,one may use one denoiser for a subset of the receivers (as assumed inFIG. 5), or one central denoiser (FIG. 6), acting for all receivers. Infact, many possible configurations may be contemplated, ranging from afew-to-one mapping between receivers and denoisers, to a unique denoiserconnected to all receivers to denoise signals therefrom, as illustratedin FIGS. 4-6.

For example, in the embodiment of FIG. 4, the system is a radiointerferometry system 4 and the receivers are radio astronomy antennas410, which are grouped in stations 420 in the system 4. In such a case,the system 4 preferably comprises (at least) one denoiser 430 perstation, connected to each of the receivers of a station. It may,however, be preferable to provide more than one denoiser per station ifantennas are scattered, geographically, even within a same station.

In all cases, the ordering S12 of the time-series is performed based onthe different locations of the antennas 410, according to principlealready discussed in reference to FIG. 1. Yet, and contrary to the priorart approaches, the noise reduction is performed at antenna-level here,i.e., prior to beamforming. In that respect, additional beamformingoperations may be involved, as usual in interferometry applications. Inparticular, the denoised, spatially phased series as obtained at the endof step S16 or step S26 may be subsequently summed (step S31, FIG. 3),at the level of each station 420, using beamforming techniques, toobtain beamformed outputs. Then, correlation values may then be obtainedS32, according to the beamformed outputs obtained at step S31. Steps S31may be performed by one or more entities 450.

Referring now to FIG. 5, in embodiments, the system is a MRI system 5and the receivers are radiofrequency coils 510 of this MRI system 5. Thepresent methods may indeed be applied to signals received from a set ofradiofrequency coils 510, 511 (only one such set is depicted in FIG. 5,for simplicity). Similarly, the system 5 may be a nuclear magneticresonance (NMR) spectroscopy system. In MRI or NMR applications, amagnetic resonance (MR) transceiver 521-522 typically generates 521wideband excitation signals that are sent to one or more excitationcoils 511 in consecutive measurement bursts, and processes signalsreceived 522 after each burst from one or more receiving coils 510 todetect narrowband signals at the output of subchannels of a filter bank524, after analog-to-digital conversion 523. The position of the magnet514 is usually modified after each measurement to differentiate thespectral contributions to the received signal from each volume element.Thus, a denoiser 530 may here be implemented as part of the filter bankor after a classical bank 524, as assumed in FIG. 5. The denoiser will,in all cases, operate on series obtained S13 by re-ordering S12 timeseries received S11 from two or more receiving coils.

In other embodiments, the present methods may be applied to signalsreceived from arrays of transducers 610 of an ultrasound apparatus 6 andthe present systems may include such an apparatus 6. The signalsreceived may be processed by a denoiser 630, e.g., in a central fashion.Yet, depending on the number and locations of the transducers 610,several denoising units may be desired. More generally, the presentmethods and systems may be directed to a sensor network.

Finally, the present invention may be implemented as a computer programproduct. This will be discussed in detail in sect. 5.

The above embodiments have been succinctly described in reference to theaccompanying drawings and may accommodate a number of variants. Severalcombinations of the above features may be contemplated. Examples aregiven in the next sections.

2. Modified Singular Spectrum Analysis Techniques Used in Embodiments

Prior art denoising methods are typically only able to handle low noiserelative to signal power. In contrast to such approaches, embodiments ofthe present invention can be applied to remove high noise fromphased-array signals at a specific time-instance.

Ref [1], see the background section, describes two non-parametric andmodel-free methods commonly used to reduce the noise from observations(i.e., the so called Cadzow's basic algorithm and the basic SingularSpectrum Analysis, or SSA). Both methods are closely related toalternating projections. The main assumption behind both methods is thatthe series can be represented as a sum of different components such astrend, harmonics and noise, see also ref. [2]. These methods performnoise reduction in any series under high signal-to-noise ratio.

Generally speaking, methods based on singular value decomposition (SVD)do not offer the possibility to remove high noise in the series. Inaddition, the existing methods for noise reduction in phased-arrays doesnot offer very accurate results.

As discussed in the previous section, embodiments of the presentinvention involve SVD-like techniques to significantly reduce the noisein phased-arrays in presence of very high instrumental noise. Suchtechniques are now discussed in more detail.

The true phased-array signal shape usually disappears in presence ofinstrument noise, see FIG. 7A. Yet, embodiments of the present inventionprovide a solution to significantly reduce the noise in the observedphased-array signal with high accuracy. This can be achieved byexploiting two properties of such systems. As it may be realized: (a)the instrument noise is very high relative to the true phased-arraysignal; and (b) the true phased-array signal is slowly varying.

Such embodiments employ an SSA technique so as to split the observedphased-array signal into true signal and noise components. Theunderlying methods are thus closely associated to the structuredlow-rank approximation problem. They include particular filteringoperations in singular spectrum of a suitably chosen projection space.

In reference to FIG. 2, the modified SSA steps used can be thefollowing:

Embedding, step S21 forms a trajectory matrix {tilde over (X)} of theobserved phased-array signal {tilde over (x)};

Singular Value Decomposition; step S22 sorts singular-triples (u_(n),σ_(n), v_(n)) for n∈{1, . . . , L} from the largest singular-value tothe lowest, i.e., σ₁≥ . . . ≥σ_(L)≥0;

Grouping: step S23 partitions the set of singular-triples

. A threshold τ∈

⁺ is set on singular-values and the singular-triples are split into twodisjoint sets

₁ and

₂ such that ∀a∈

₁ and b∈

₂, σ_(a)≥σ_(b);

Eliminating high frequencies: step S24 analyses and filters in thefrequency domain. A threshold ω∈[0, 2π) is set on the frequency band anda low-pass filter applied with cutoff frequency ω to u_(n) and v_(n)∀n∈

₂. The filtered singular-vectors are denoted by û_(n) and {circumflexover (v)}_(n);

Decomposition (low-rank approximation): step S25 approximate thetrajectory matrix {circumflex over (X)} of the true signal by Σ_(b∈)

₂ û_(b)σ_(b){circumflex over (v)}_(b) ^(T); and

Diagonal averaging: step S26 reconstructs the approximated de-noisedseries {circumflex over (x)} by applying the inverse of the embeddingoperation to {circumflex over (X)}.

3. Properties

Roughly speaking, the SSA technique presented in the previous sectionseparates components associated with the true signal and componentsassociated with noise. It is all the more suited if the followingconditions are fulfilled:

The noise should be high relative to the true signal power;

The noise and the true signal should be approximately separable; and

The noise and the true signal should be distinguishable.

The first of these conditions is an additional necessary condition setfor distinguishability, while the other two conditions follow from thesame discussion as in basic SSA.

Let {tilde over (x)}_(j) denote the ordered sequence of samples asobtained by ordering time-series from a set of receivers at step S13, ata specific time t_(j). Such receivers may for instance be closelylocated antennas. Thus, we have:{tilde over (x)} _(j) =x _(j) +n _(j),  (1)where x_(j) and n_(j) correspond to the sequence and the correspondingnoise sequence at t_(j), respectively. Generally, thermal noise isstatistically independent from the true signal and among differentreceivers, which in turn implies that noise and true signal areapproximately separable.

As implied before, low signal-to-noise ratio and slow-varying propertyof the true signal are significantly prior information in the presentmethod. Both properties impact the distinguishability requirement setforth above. We exploit high thermal noise property to eliminate

₁ since the high valued singular-values set belongs to the noisecomponent. Note that

₂ includes true signal and high frequency components of the noisefollowings:

₁ is leading to

₂ by definition; and

Trend is in leading singular-triples.

Hence, eliminating high frequencies in singular-vectors in

₂ followed by the steps S25 (decomposition) and S26 (diagonal averaging)will give the trend of the true signal, which may, in turn, even beequivalent to the true signal, owing to the slow-varying property of thetrue signal.

Finally, we note that the approximate separability condition still holdsdue to independence between noise and true signal.

4. Detailed Description of Particular Embodiments 4.1 Radio Astronomy

Modern radio telescopes correlate the signals measured by thousands ofantennas at various locations on the ground, in order to infer an imageof the sky. To reduce the amount of data sent to a central processor andincrease the signal-to-noise ratio, closely-located antennas are groupedtogether in stations (FIG. 4) and beamformed (FIG. 3). One of the mainchallenges that radio astronomy interferometers faces is the very highinstrument noise relative to signal power, which, in turn, leads tonoisy images. The usual technique for recovery of the sky image is toproduce many correlator outputs to overcome the noise, and to use thoseoutputs, after calibration, to ascertain an image. However, noisereduction has not been processed at antenna level so far. To cope withthe non-coherent thermal noise introduced by the antennas, embodimentsof the invention offer an efficient and accurate method of noisereduction.

Thus, a preferred embodiment in radio astronomy is, for a givencollection of antennas, to:

-   -   Order S12 the series of observations taken S11 from antennas        geographically, at each time-instance, to obtain S13        spatially-phased series;    -   Pass these through the denoising filter described above, steps        S21-S26 (S14-S16); and    -   Reconstruct S17 the desired signal by:        -   Summing S31 de-noised series at station level by            beamforming; and        -   Producing S32 correlation values from the beamformed            outputs.

FIGS. 7 and 8 illustrate the application of embodiments of the inventionin radio astronomy on sky images. A case study on the performance ofsuch embodiments applied to antenna observations is given in FIG. 1.Namely, FIGS. 7A and 7B show series of observations taken fromgeographically close antennas. The above method is applied to antennaobservations within each station (dashed line, FIG. 7B). FIG. 7Acompares antenna observations that include thermal noise with antennaobservations when no thermal noise is present (as obtained from asynthetic data set). FIG. 7B demonstrates that the noise can besignificantly reduced.

FIGS. 8A and 8B represent differences (as density plots) between a truesignal (without thermal noise) of a sky image and, on the one hand, anestimate thereof as reconstructed without denoising at all (FIG. 8A)and, on the other hand, an estimate as reconstructed according to anembodiment (FIG. 8B). It is clear that the denoising obtained accordingto such an embodiment dramatically improves performance.

4.2 Medical Imaging

Ultrasound consists of arrays of transducers. When the data is receivedby the transducers, the resultant image is very noisy. Embodiments ofthe invention can help to remove this noise. The potential benefits ofsuch embodiments for ultrasound applications include more accurateimages (noise is removed) and a lower number of necessary transducers,which, in turn, enable low-cost and portable devices.

A preferred embodiment of the present methods for denoising thetime-series received from the transducers, comprises the following steps(FIGS. 1, 2 and 6):

Pick an appropriate ordering of the transducers 610, S12, to obtain S13slowly-varying, spatially phased series; and

At each time instant, pass the digitally sampled transducer values ofthe spatially phased series through the denoiser 630, steps S21-S26(S14-S16); and

Pass the denoised values for further processing, to reconstruct S17 thedesired signal.

MRI systems are moving to using more and more coils in parallel, therebyacting as a phased-array [3]. For a given set of receiving coils,preferred embodiments follow a procedure similar to that used in theultrasound case.

4.3 General Time-Series Observing Sensor Networks

A sensor network such as those used in “smart-cities” initiatives formonitoring water pressure, temperature, etc., can be regarded as aphased array, whose data would benefit from denoising methods asdiscussed in section 1. In correlating devices connected in an“internet-of-things”, such methods may provide crucial benefits bydenoising the numerous signals collected.

5. Technical Implementation Details 5.1 Computerized Systems and Devices

Computerized systems and devices can be suitably designed forimplementing embodiments of the present invention as described herein.In that respect, it can be appreciated that the methods described hereinare largely non-interactive and automated. In exemplary embodiments, themethods described herein can be implemented either in an interactive,partly-interactive or non-interactive system. The methods describedherein can be implemented in software, hardware, or a combinationthereof. In exemplary embodiments, the methods described herein areimplemented in software, as an executable program, the latter executedby suitable digital processing devices. More generally, embodiments ofthe present invention can be implemented wherein general-purpose digitalcomputers, such as personal computers, workstations, etc., are used.

For instance, the system 100 depicted in FIG. 9 schematically representsa computerized unit 101, e.g., a general- or specific-purpose computer,which may be used in place or as part of the devices 430, 450, 530, 630described earlier. As such, the unit 101 may interact with receivers (ortransceivers, transducers) 410, 510, 610, e.g., via converters and I/Ounits 145-155.

In exemplary embodiments, in terms of hardware architecture, as shown inFIG. 9, the unit 101 includes a processor 105, memory 110 coupled to amemory controller 115. One or more input and/or output (I/O) devices145, 150, 155 (or peripherals) are communicatively coupled via a localinput/output controller 135. The input/output controller 135 can becoupled to or include one or more buses and a system bus 140, as knownin the art. The input/output controller 135 may have additionalelements, which are omitted for simplicity, such as controllers, buffers(caches), drivers, repeaters, and receivers, to enable communications.Further, the local interface may include address, control, and/or dataconnections to enable appropriate communications among theaforementioned components.

The processor 105 is a hardware device for executing software,particularly that stored in memory 110. The processor 105 can be anycustom made or commercially available processor, a central processingunit (CPU), an auxiliary processor among several processors associatedwith the computer 101, a semiconductor based microprocessor (in the formof a microchip or chip set), or generally any device for executingsoftware instructions.

The memory 110 can include any one or combination of volatile memoryelements (e.g., random access memory) and nonvolatile memory elements.Moreover, the memory 110 may incorporate electronic, magnetic, optical,and/or other types of storage media. Note that the memory 110 can have adistributed architecture, where various components are situated remotefrom one another, but can be accessed by the processor 105.

The software in memory 110 may include one or more separate programs,each of which comprises an ordered listing of executable instructionsfor implementing logical functions. In the example of FIG. 9, thesoftware in the memory 110 includes methods described herein inaccordance with exemplary embodiments and, in particular, a suitableoperating system (OS) 111. The OS 111 essentially controls the executionof other computer programs and provides scheduling, input-outputcontrol, file and data management, memory management, and communicationcontrol and related services.

The methods described herein may be in the form of a source program,executable program (object code), script, or any other entity comprisinga set of instructions to be performed. When in a source program form,then the program needs to be translated via a compiler, assembler,interpreter, or the like, as known per se, which may or may not beincluded within the memory 110, so as to operate properly in connectionwith the OS 111. Furthermore, the methods can be written as an objectoriented programming language, which has classes of data and methods, ora procedure programming language, which has routines, subroutines,and/or functions.

Possibly, a conventional keyboard and mouse can be coupled to theinput/output controller 135. Other I/O devices 140-155 may include or beconnected to other hardware devices 10, as noted earlier.

In addition, the I/O devices 140-155 may further include or be connectedto devices 410, 510, 610 that communicate outputs, e.g., time series.The system 100 can further include a display controller 125 coupled to adisplay 130. In exemplary embodiments, the system 100 can furtherinclude a network interface or transceiver 160 for coupling to a network165, to enable, in turn, data communication to/from other, externalcomponents.

The network 165 transmits and receives data between the unit 101 andexternal systems. The network 165 is possibly implemented in a wirelessfashion, e.g., using wireless protocols and technologies, such as Wifi,WiMax, etc. The network 165 may be a fixed wireless network, a wirelesslocal area network (LAN), a wireless wide area network (WAN) a personalarea network (PAN), a virtual private network (VPN), intranet or othersuitable network system and includes equipment for receiving andtransmitting signals.

The network 165 can also be an IP-based network for communicationbetween the unit 101 and any external server, client and the like via abroadband connection. In exemplary embodiments, network 165 can be amanaged IP network administered by a service provider. Besides, thenetwork 165 can be a packet-switched network such as a LAN, WAN,Internet network, an Internet of things network, etc.

If the unit 101 is a PC, workstation, intelligent device or the like,the software in the memory 110 may further include a basic input outputsystem (BIOS). The BIOS is stored in ROM so that the BIOS can beexecuted when the computer 101 is activated. When the unit 101 is inoperation, the processor 105 is configured to execute software storedwithin the memory 110, to communicate data to and from the memory 110,and to generally control operations of the computer 101 pursuant to thesoftware.

The methods described herein and the OS 111, in whole or in part areread by the processor 105, typically buffered within the processor 105,and then executed. When the methods described herein are implemented insoftware, the methods can be stored on any computer readable medium,such as storage 120, for use by or in connection with any computerrelated system or method.

5.2 Computer Program Products

The present invention may be an apparatus, a method, and/or a computerprogram product. The computer program product may include a computerreadable storage medium (or media) having computer readable programinstructions thereon for causing a processor to carry out aspects of thepresent invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, or either source code or object code written in anycombination of one or more programming languages, including an objectoriented programming language such as Smalltalk, C++ or the like, andconventional procedural programming languages, such as the C programminglanguage or similar programming languages. The computer readable programinstructions may execute entirely on the user's computer, partly on theuser's computer, as a stand-alone software package, partly on the user'scomputer and partly on a remote computer or entirely on the remotecomputer or server. In the latter scenario, the remote computer may beconnected to the user's computer through any type of network, includinga local area network (LAN) or a wide area network (WAN), or theconnection may be made to an external computer (for example, through theInternet using an Internet Service Provider). In some embodiments,electronic circuitry including, for example, programmable logiccircuitry, field-programmable gate arrays (FPGA), or programmable logicarrays (PLA) may execute the computer readable program instructions byutilizing state information of the computer readable programinstructions to personalize the electronic circuitry, in order toperform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the block may occur out of theorder noted in the figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

While the present invention has been described with reference to alimited number of embodiments, variants and the accompanying drawings,it will be understood by those skilled in the art that various changesmay be made and equivalents may be substituted without departing fromthe scope of the present invention. In particular, a feature(device-like or method-like) recited in a given embodiment, variant orshown in a drawing may be combined with or replace another feature inanother embodiment, variant or drawing, without departing from the scopeof the present invention. Various combinations of the features describedin respect of any of the above embodiments or variants may accordinglybe contemplated, that remain within the scope of the appended claims. Inaddition, many minor modifications may be made to adapt a particularsituation or material to the teachings of the present invention withoutdeparting from its scope. Therefore, it is intended that the presentinvention not be limited to the particular embodiments disclosed, butthat the present invention will include all embodiments falling withinthe scope of the appended claims. In addition, many other variants thanexplicitly touched above can be contemplated. For example, otherapplications than those explicitly mentioned may benefit from denoisingmethods as described herein.

What is claimed is:
 1. A computerized method to reduce noise inphased-array signals from a set of radio frequency coils of a magneticresonance imaging system, the coils being located at differentlocations, wherein the method comprises: receiving time-series from theradiofrequency coils, the time-series forming phased-array signals;ordering the time-series based on the different locations of theradiofrequency coils and obtaining, from the ordered time-series,spatially phased series, wherein each of the spatially phased seriesobtained comprises a series of signal values that are spatially phased;identifying a noise component in each of the spatially phased seriesobtained and removing the identified noise component from said each ofthe spatially phased series obtained to obtain denoised, spatiallyphased series; and passing the denoised, spatially phased series forfurther processing to reconstruct an estimate of a true signal accordingto each of the denoised, spatially phased series obtained, wherein: saidset of radiofrequency coils are arranged in an array; saidradiofrequency coils introduce noise; and said ordering the time-seriesis performed based on locations of the radiofrequency coils.
 2. Themethod according to claim 1, wherein: said ordering the time-seriescomprises selecting time-series elements corresponding to two or moretime instances and ordering the time-series elements selected based onthe different locations of the radiofrequency coils, so that each of thespatially phased series obtained from the ordered time-series comprisesa series of signal values that correspond to signal sensed two or moretime instances.
 3. The method according to claim 1, wherein: saidordering the time-series is performed for each time-instance of thetime-series received, based on the different locations of theradiofrequency coils, so that each of the spatially phased seriesobtained for each time-instance comprises a series of signal values thatcorrespond to signal sensed at a same time instance.
 4. The methodaccording to claim 3, wherein: said ordering the time-series comprisesordering the time-series based on distances between the differentlocations of the radiofrequency coils.
 5. The method according to claim4, wherein: said ordering the time-series is performed by identifying asequence of radiofrequency coils, wherein an n+1th radiofrequency coilof the sequence is the closest radiofrequency coil from an n^(th)radiofrequency coil of the sequence, so as to minimize distances betweenlocations corresponding to contiguous pairs of elements in each of thespatially phased series obtained.
 6. The method according to claim 1,wherein: said identifying a noise component and removing the identifiednoise component to obtain denoised, spatially phased series is performedaccording to steps of a singular spectrum analysis.
 7. The methodaccording to claim 6, wherein: said steps of the singular spectrumanalysis include a step of eliminating frequencies above a giventhreshold in singular-vectors of one of two disjoint sets of singulartriples that comprises lowest singular-values, prior to reconstruct anapproximate de-noised, spatially phased series.
 8. A computerized methodto reduce noise in phased-array signals from a set of transducers of anultrasound apparatus, the transducers being at different locations,wherein the method comprises: receiving time-series from thetransducers, the time-series forming phased-array signals; ordering thetime-series based on the different locations of the transducers andobtaining, from the ordered time-series, spatially phased series,wherein each of the spatially phased series obtained comprises a seriesof signal values that are spatially phased; and identifying a noisecomponent in each of the spatially phased series obtained and removingthe identified noise component from said each of the spatially phasedseries obtained to obtain denoised, spatially phased series; and passingthe denoised, spatially phased series for further processing toreconstruct an estimate of a true signal according to each of thedenoised, spatially phased series obtained, wherein: said set ofradiofrequency coils are arranged in an array; said radiofrequency coilsintroduce noise; and said ordering the time-series is performed based onthe different locations of the radiofrequency coils.
 9. A computerizedmethod to reduce noise in phased-array signals from a set ofInternet-of-things sensors of a sensor network, the sensors being atdifferent locations, wherein the method comprises: receiving time-seriesfrom the sensors, the time-series forming phased-array signals; orderingthe time-series based on the different locations of the sensors andobtaining, from the ordered time-series, spatially phased series,wherein each of the spatially phased series obtained comprises a seriesof signal values that are spatially phased; and identifying a noisecomponent in each of the spatially phased series obtained and removingthe identified noise component from said each of the spatially phasedseries obtained to obtain denoised, spatially phased series; and passingthe denoised, spatially phased series for further processing toreconstruct an estimate of a true signal according to each of thedenoised, spatially phased series obtained, wherein: said set ofradiofrequency coils are arranged in an array; said radiofrequency coilsintroduce noise; and said ordering the time-series is performed based onthe different locations of the radiofrequency coils.